Answer:
[tex]f(n) = -\frac{8}{3}n[/tex]
Step-by-step explanation:
Given
[tex]-2\frac{2}{3}, -5\frac{1}{3}, -10\frac{2}{3}, -21\frac{1}{3}, -42\frac{2}{3}[/tex]
Required
The explicit formula
The above sequence is an arithmetic sequence and it is bounded by:
[tex]f(n) = a + (n - 1)d[/tex]
Where
[tex]a = -2\frac{2}{3}[/tex] -- the first term
[tex]d = f(2) -f(1)[/tex]
So, we have:
[tex]d = -5\frac{1}{3} - -2\frac{2}{3}[/tex]
[tex]d = -5\frac{1}{3} +2\frac{2}{3}[/tex]
Express as improper fraction
[tex]d = -\frac{16}{3} +\frac{8}{3}[/tex]
Take LCM
[tex]d = \frac{-16+8}{3}[/tex]
[tex]d = -\frac{8}{3}[/tex]
So, we have:
[tex]f(n) = a + (n - 1)d[/tex]
[tex]f(n) = -2\frac{2}{3} + (n - 1) * -\frac{8}{3}[/tex]
Express all fractions as improper
[tex]f(n) = -\frac{8}{3} + (n - 1) * -\frac{8}{3}[/tex]
Open brackets
[tex]f(n) = -\frac{8}{3} -\frac{8}{3}n +\frac{8}{3}[/tex]
Collect like terms
[tex]f(n) = -\frac{8}{3}n +\frac{8}{3}-\frac{8}{3}[/tex]
[tex]f(n) = -\frac{8}{3}n[/tex]
